Stent grafts and covered stents are widely used to treat aneurysms, occluded vessels, and restenosis. These devices are generally constructed from a stent and a graft material secured to the stent. In existing stent grafts and covered stents, the graft material is generally secured to the stent by suturing, taping, or stapling. However, these methods require the addition of extra material (e.g., suture thread) and thereby increase the profile of the device and the size of the sheath required to deliver the device. It has been a challenge to develop a method for securing a graft material to a stent without increasing the overall profile of the stent graft device.